Rigid

In mathematics, suppose "C" is a collection of mathematical objects (for instance sets or functions). Then we say that "C" is rigid if every "c" $in$ "C" is uniquely determined by less information about "c" than one would expect.

It should be emphasized that the above statement does not define a mathematical property. Instead, it describes in what sense the adjective rigid is typically used in mathematics, by mathematicians.

Some examples include:
#Harmonic functions on the unit disk are rigid in the sense that they are uniquely determined by their boundary values.
#Holomorphic functions are determined by the set of all derivatives at a single point. A smooth function from the real line to the complex plane is not, in general, determined by all its derivatives at a single point, but it is if we require additionally that it be possible to extend the function to one on a neighbourhood of the real line in the complex plane. The Schwarz lemma is an example of such a rigidity theorem.
#By the fundamental theorem of algebra, polynomials in C are rigid in the sense that any polynomial is completely determined by its values on any infinite set, say N, or the unit disk. Note that by the previous example, a polynomial is also determined within the set of holomorphic functions by the finite set of its non-zero derivatives at any single point.
#Linear maps L("X","Y") between vector spaces "X", "Y" are rigid in the sense that any L $in$ L("X","Y") is completely determined by its values on any set of basis vectors of "X".
#Mostow's rigidity theorem, which states that negatively curved manifolds are isomorphic if some rather weak conditions on them hold.
#A well-ordered set is rigid in the sense that the only (order-preserving) automorphism on it is the identity function. Consequently, an isomorphism between two given well-ordered sets will be unique.
#A rigid motion of a subset of Euclidean space is not always defined the same: it may be any distance-preserving transformation of the collection of points (i.e. a composition of translations, rotations, and reflections), or only those preserving orientation (i.e. a composition of translations and rotations). In the latter case the concept of rigidity is analogous to that of a physically inflexible solid, which must be moved as a single entity so that its movement (up to atomic motions indiscernible to the naked eye) is completely determined by the displacement of a single "point" and the orientation of the solid body about that point. More generally, a rigid motion of a metric space is a (self)-isometry.